Essential Concepts

Finding Limits

A limit describes the behavior of a function as the input values get close to a specific value. We write [latex]\underset{x\to a}{\mathrm{lim}}f(x)=L[/latex] to indicate that as [latex]x[/latex] approaches [latex]a[/latex], the output value [latex]f(x)[/latex] approaches [latex]L[/latex].

The limit of a function as [latex]x[/latex] approaches [latex]a[/latex] can exist even when [latex]f(a)[/latex] does not exist or when [latex]f(a)\neq L[/latex]. The value of the limit is not affected by the output value of [latex]f(x)[/latex] at [latex]a[/latex].

One-sided limits:

• Left-hand limit: [latex]\underset{x\to a^{-}}{\mathrm{lim}}f(x)=L[/latex] means the output approaches [latex]L[/latex] as [latex]x[/latex] approaches [latex]a[/latex] from the left (where [latex]x
• Right-hand limit: [latex]\underset{x\to a^{+}}{\mathrm{lim}}f(x)=L[/latex] means the output approaches [latex]L[/latex] as [latex]x[/latex] approaches [latex]a[/latex] from the right (where [latex]x>a[/latex] and [latex]x\neq a[/latex])

Two-sided limits: A two-sided limit exists if and only if both one-sided limits exist and are equal: [latex]\underset{x\to a^{-}}{\mathrm{lim}}f(x)=\underset{x\to a^{+}}{\mathrm{lim}}f(x)[/latex]. When we refer to “a limit” without specifying, we mean a two-sided limit.

Finding limits using graphs: A graph provides a visual method for determining limits. If the function has a limit as [latex]x[/latex] approaches [latex]a[/latex], the branches of the graph will approach the same y-coordinate near [latex]x=a[/latex] from both the left and right.

1. Examine the graph as [latex]x[/latex] approaches [latex]a[/latex] from the left to determine if a left-hand limit exists
2. Examine the graph as [latex]x[/latex] approaches [latex]a[/latex] from the right to determine if a right-hand limit exists
3. If both one-sided limits exist and are equal, then the two-sided limit exists
4. Check if there is a point at [latex]x=a[/latex] to determine if [latex]f(a)[/latex] exists

Finding limits using tables: A table can be used to determine if a function has a limit by showing input values that approach [latex]a[/latex] from both directions.

1. Choose input values that approach [latex]a[/latex] from both the left and right
2. Evaluate the function at each input value and record in a table
3. Determine if the output values approach a specific number [latex]L[/latex] from both sides
4. If the output values approach the same number from both directions, the limit is [latex]L[/latex]

Properties of Limits

When limits exist and are finite, we can perform operations using the properties below. Let [latex]\underset{x\to a}{\mathrm{lim}}f(x)=A[/latex] and [latex]\underset{x\to a}{\mathrm{lim}}g(x)=B[/latex].

Basic properties:

• Constant: [latex]\underset{x\to a}{\mathrm{lim}}k=k[/latex]
• Constant times a function: [latex]\underset{x\to a}{\mathrm{lim}}[k\cdot f(x)]=k\cdot A[/latex]
• Sum: [latex]\underset{x\to a}{\mathrm{lim}}[f(x)+g(x)]=A+B[/latex]
• Difference: [latex]\underset{x\to a}{\mathrm{lim}}[f(x)-g(x)]=A-B[/latex]
• Product: [latex]\underset{x\to a}{\mathrm{lim}}[f(x)\cdot g(x)]=A\cdot B[/latex]
• Quotient: [latex]\underset{x\to a}{\mathrm{lim}}\frac{f(x)}{g(x)}=\frac{A}{B}[/latex], where [latex]B\neq 0[/latex]
• Power: [latex]\underset{x\to a}{\mathrm{lim}}[f(x)]^{n}=A^{n}[/latex], where [latex]n[/latex] is a positive integer
• Root: [latex]\underset{x\to a}{\mathrm{lim}}\sqrt[n]{f(x)}=\sqrt[n]{A}[/latex], where [latex]n[/latex] is a positive integer

Limits of polynomials: For a polynomial function [latex]p(x)[/latex], we have [latex]\underset{x\to a}{\mathrm{lim}}p(x)=p(a)[/latex]. The limit can be found by finding the sum of the limits of the individual terms, or more simply, by direct substitution of [latex]a[/latex] into the polynomial.

Evaluating limits of quotients: When the denominator evaluates to 0, rewrite the function algebraically before applying limit properties:

1. Factor the numerator and denominator, then cancel common factors and simplify
2. Find the LCD if the numerator or denominator contains complex fractions, then simplify
3. Multiply by the conjugate if the expression contains a square root; multiply numerator and denominator by the conjugate
4. Use numeric evidence (tables or graphs) for functions with absolute values, or set up piecewise

Continuity

A function is continuous at [latex]x=a[/latex] if there are no holes or breaks in its graph at that point. A discontinuous function is one that has any hole or break in its graph.

Three conditions for continuity: A function [latex]f(x)[/latex] is continuous at [latex]x=a[/latex] if all three conditions hold:

1. Condition 1: [latex]f(a)[/latex] exists
2. Condition 2: [latex]\underset{x\to a}{\mathrm{lim}}f(x)[/latex] exists at [latex]x=a[/latex]
3. Condition 3: [latex]\underset{x\to a}{\mathrm{lim}}f(x)=f(a)[/latex]

If any condition fails, the function is discontinuous at [latex]x=a[/latex].

Functions that are continuous everywhere:

• Polynomial functions
• Exponential functions
• Sine functions
• Cosine functions

Functions continuous on their domain:

• Logarithmic functions
• Tangent functions
• Rational functions

Types of discontinuities:

• Jump discontinuity: Occurs when left-hand and right-hand limits both exist but are not equal: [latex]\underset{x\to a^{-}}{\mathrm{lim}}f(x)\neq\underset{x\to a^{+}}{\mathrm{lim}}f(x)[/latex]. The graph “jumps” from one value to another at the point of discontinuity.
• Removable discontinuity: Occurs when [latex]\underset{x\to a}{\mathrm{lim}}f(x)[/latex] exists, but either [latex]f(a)[/latex] does not exist or [latex]f(a)\neq\underset{x\to a}{\mathrm{lim}}f(x)[/latex]. There is a “hole” in the graph. The function can be redefined at its discontinuous point to make it continuous.
• Infinite discontinuity: Occurs when one or both of the one-sided limits approaches positive or negative infinity. The graph typically has a vertical asymptote at [latex]x=a[/latex].

Determining continuity of piecewise functions:

To check continuity at a boundary point [latex]x=a[/latex]:

1. Verify each component function is continuous on its domain
2. Evaluate the left-hand limit using the piece defined for [latex]x
3. Evaluate the right-hand limit using the piece defined for [latex]x>a[/latex] (or [latex]x\geq a[/latex])
4. Find [latex]f(a)[/latex] using whichever piece includes [latex]x=a[/latex] in its domain
5. Check if all three conditions of continuity are satisfied: [latex]f(a)[/latex] exists, [latex]\underset{x\to a}{\mathrm{lim}}f(x)[/latex] exists (both one-sided limits are equal), and [latex]\underset{x\to a}{\mathrm{lim}}f(x)=f(a)[/latex]

Derivatives

The derivative of a function at a point represents the instantaneous rate of change at that point. It can also be interpreted as the slope of the tangent line to the graph at that point.

Instantaneous rate of change: The derivative measures how quickly the output of a function is changing at a specific input value, unlike average rate of change which measures change over an interval.

Tangent line: A line that touches the graph of a function at exactly one point and has the same slope as the function at that point. The equation of the tangent line to [latex]f(x)[/latex] at [latex]x=a[/latex] can be found using the derivative and point-slope form.

Key Equations

Limit notation	[latex]\underset{x\to a}{\mathrm{lim}}f(x)=L[/latex]
Left-hand limit	[latex]\underset{x\to a^{-}}{\mathrm{lim}}f(x)=L[/latex]
Right-hand limit	[latex]\underset{x\to a^{+}}{\mathrm{lim}}f(x)=L[/latex]
Limit of a constant	[latex]\underset{x\to a}{\mathrm{lim}}k=k[/latex]
Limit of constant times a function	[latex]\underset{x\to a}{\mathrm{lim}}[k\cdot f(x)]=k\underset{x\to a}{\mathrm{lim}}f(x)[/latex]
Limit of a sum	[latex]\underset{x\to a}{\mathrm{lim}}[f(x)+g(x)]=\underset{x\to a}{\mathrm{lim}}f(x)+\underset{x\to a}{\mathrm{lim}}g(x)[/latex]
Limit of a difference	[latex]\underset{x\to a}{\mathrm{lim}}[f(x)-g(x)]=\underset{x\to a}{\mathrm{lim}}f(x)-\underset{x\to a}{\mathrm{lim}}g(x)[/latex]
Limit of a product	[latex]\underset{x\to a}{\mathrm{lim}}[f(x)\cdot g(x)]=\underset{x\to a}{\mathrm{lim}}f(x)\cdot\underset{x\to a}{\mathrm{lim}}g(x)[/latex]
Limit of a quotient	[latex]\underset{x\to a}{\mathrm{lim}}\frac{f(x)}{g(x)}=\frac{\underset{x\to a}{\mathrm{lim}}f(x)}{\underset{x\to a}{\mathrm{lim}}g(x)}[/latex], where [latex]\underset{x\to a}{\mathrm{lim}}g(x)\neq 0[/latex]
Limit of a power	[latex]\underset{x\to a}{\mathrm{lim}}[f(x)]^{n}=[\underset{x\to a}{\mathrm{lim}}f(x)]^{n}[/latex]
Limit of a root	[latex]\underset{x\to a}{\mathrm{lim}}\sqrt[n]{f(x)}=\sqrt[n]{\underset{x\to a}{\mathrm{lim}}f(x)}[/latex]
Limit of a polynomial	[latex]\underset{x\to a}{\mathrm{lim}}p(x)=p(a)[/latex]

Glossary

continuous function

A function that has no holes or breaks in its graph; a function [latex]f(x)[/latex] is continuous at [latex]x=a[/latex] if [latex]f(a)[/latex] exists, [latex]\underset{x\to a}{\mathrm{lim}}f(x)[/latex] exists, and [latex]\underset{x\to a}{\mathrm{lim}}f(x)=f(a)[/latex].

derivative

The instantaneous rate of change of a function at a point; also represents the slope of the tangent line to the graph of the function at that point.

discontinuous function

A function that is not continuous at [latex]x=a[/latex]; a function that has one or more holes or breaks in its graph.

infinite discontinuity

A point of discontinuity in a function [latex]f(x)[/latex] at [latex]x=a[/latex] where one or both of the one-sided limits approaches positive or negative infinity; the graph typically has a vertical asymptote at [latex]x=a[/latex].

instantaneous rate of change

The rate at which a function is changing at a specific point, measured by the derivative; describes how quickly the output changes relative to the input at an exact moment.

jump discontinuity

A point of discontinuity in a function [latex]f(x)[/latex] at [latex]x=a[/latex] where both the left-hand and right-hand limits exist, but [latex]\underset{x\to a^{-}}{\mathrm{lim}}f(x)\neq\underset{x\to a^{+}}{\mathrm{lim}}f(x)[/latex]; the graph “jumps” from one value to another.

left-hand limit

The limit of values of [latex]f(x)[/latex] as [latex]x[/latex] approaches [latex]a[/latex] from the left, denoted [latex]\underset{x\to a^{-}}{\mathrm{lim}}f(x)=L[/latex]. The values of [latex]f(x)[/latex] can get as close to the limit [latex]L[/latex] as we like by taking values of [latex]x[/latex] sufficiently close to [latex]a[/latex] such that [latex]x

limit

When it exists, the value [latex]L[/latex] that the output of a function [latex]f(x)[/latex] approaches as the input [latex]x[/latex] gets closer and closer to [latex]a[/latex] but does not equal [latex]a[/latex]. The value of the output [latex]f(x)[/latex] can get as close to [latex]L[/latex] as we choose by using input values of [latex]x[/latex] sufficiently near to [latex]x=a[/latex], but not necessarily at [latex]x=a[/latex]. Denoted [latex]\underset{x\to a}{\mathrm{lim}}f(x)=L[/latex].

properties of limits

A collection of theorems for finding limits of functions by performing mathematical operations on the limits.

removable discontinuity

A point of discontinuity in a function [latex]f(x)[/latex] at [latex]x=a[/latex] where the limit exists but either [latex]f(a)[/latex] does not exist or [latex]f(a)\neq\underset{x\to a}{\mathrm{lim}}f(x)[/latex]; the function can be redefined at its discontinuous point to make it continuous.

right-hand limit

The limit of values of [latex]f(x)[/latex] as [latex]x[/latex] approaches [latex]a[/latex] from the right, denoted [latex]\underset{x\to a^{+}}{\mathrm{lim}}f(x)=L[/latex]. The values of [latex]f(x)[/latex] can get as close to the limit [latex]L[/latex] as we like by taking values of [latex]x[/latex] sufficiently close to [latex]a[/latex] where [latex]x>a[/latex] and [latex]x\neq a[/latex].

stepwise function

A function that remains constant over intervals and then jumps instantaneously to different values; an example of a function with jump discontinuities.

tangent line

A line that touches the graph of a function at exactly one point and has the same instantaneous rate of change (slope) as the function at that point.

two-sided limit

The limit of a function [latex]f(x)[/latex] as [latex]x[/latex] approaches [latex]a[/latex] is equal to [latex]L[/latex], denoted [latex]\underset{x\to a}{\mathrm{lim}}f(x)=L[/latex], if and only if [latex]\underset{x\to a^{-}}{\mathrm{lim}}f(x)=\underset{x\to a^{+}}{\mathrm{lim}}f(x)[/latex].